we can assume without loss of generality that x and y are of the form x pabcq and y pbcdq,
|
|
- Moris Ronald Stokes
- 5 years ago
- Views:
Transcription
1 1. Symmetric and alternating groups. SOLUTIONS Math A4900 Homework 9 11/1/2017 (a) In class we showed that S n is generated by T tpi jq 1 ď i ă j ď nu, the set of transpositions in S n. Show by induction on j i, that for i ă j, pi jq pi i ` 1qpi ` 1 i ` 2q pj 2 j 1qpj 1 jqpj 2 j 1q pi ` 1 i ` 2qpi i ` 1q, and conclude S n is generated by T 1 tpi i`1q 1 ď i ă nu, the set of adjacent transpositions. Proof. We show the claim by induction on j i. When j i 1, the claim is trivially true. Now assume that Then pi ` 1 jq pi ` 1 i ` 2q pj 2 j 1qpj 1 jqpj 2 j 1q pi ` 1 i ` 2q. pi i`1qpi`1 i`2q pj 2 j 1qpj 1 jqpj 2 j 1q pi`1 i`2qpi i`1q IH pi i`1qpi`1 jqpi i`1q pi jq. Thus all transpositions are generated by adjacent transpositions, and so implying that S n xt 1 y. S n Ě xt 1 y Ě xt y S n, (b) Show that σ 2 is an even permutation for every σ P S n. Proof. Let ɛ : S n Ñ t 1uˆ be the sign function. Then for any σ P S n, since ɛ is a homomorphism, we have ɛpσ 2 q ɛpσq 2 p 1q 2 1. So σ 2 is even. (c) Let x, y be distinct 3-cycles in S n. (i) Set n 4 and assume x y 1. Show xx, yy A 4. Proof. Since x y 1 and xx, yy xx, y 1 y, we can assume without loss of generality that x and y are of the form where ta, b, c, du r4s. Then x pabcq and y pbcdq, 1, x, x 2 pacbq, y, y 2 pbdcq, yxy 1 pacdq, and yx 2 y 1 pcadq padcq are distinct elements of xx, yy, so that xx, yy ě 7. But since x and y are both even, we know xx, yy ď A 4, and so xx, yy divides A So xx, yy 12, and therefore xx, yy A 4. 1
2 2 (ii) Set n 5 and assume x y 1. Show that either x and y both fix some common elements of r5s (there is some i P r5s such that xpiq i and ypiq i) and xx, yy A 4, or x and y do not fix any common elements of r5s (for all i P r5s, if xpiq i then ypiq i) and xx, yy A 5. Proof. If x and y both fix some common element a P r5s, then if A r5s tau, we have x, y P S A S 4. So this case reduces to the setting of the previous part, and thus xx, yy A 4. Otherwise, using similar reasoning as in the previous case, without loss of generality, we can write x pabcq and y pcdeq, where ta, b, c, d, eu r5s. Again, 1 P xx, yy. And x, x 2, y, and x 2 are distinct 3-cycles contained in xx, yy. And we can generate all 20 3-cycles as follows, using our conjugation rule for permutations: pabcq x pacbq x 1 pcdeq y pcedq y 1 pabdq yxy 1 padbq pyxy 1q 1 pabeq y 1 xy paebq py 1xyq 1 padeq xyx 1 paedq pxyx 1q 1 pbdeq x 1 yx pbedq px 1yxq 1 paceq yxy 1x 1y 1 paecq pbcdq xyxy 1x 1 pbdcq pxyxy 1x 1q 1 pacdq x 1yx 1y 1 x padcq pbceq xy 1xyx 1 pbecq pxy 1xyx 1q 1 pyxy 1 x 1 y 1 q 1 px 1 yx 1 y 1 xq 1 Now using pairs x 1, y 1 of these three cycles that overlap at exactly two values (like x 1 pabcq and y 1 pbcdq) as in the previous part, we can also generate all ` permutations that are the product of two disjoint 2-cycles (since they are elements of the corresponding isomorphic image of A 4 ). So xx, yy ě t1u ` t3-cyclesu ` tproduct of two disjoint 2-cyclesu 1 ` 20 ` But since x and y are both even, we know xx, yy ď A 5, and so xx, yy divides A So xx, yy 60, and therefore xx, yy A 5. (iii) Show, for all n, that xx, yy is isomorphic to one of Z 3, A 4, A 5, or Z 3 ˆ Z 3. Proof. Writing x px 1 x 2 x 3 q, y py 1, y 2, y 3 q, and A tx 1, x 2, x 3, y 1, y 2, y 3 u, then we have xx, yy ď S A S A. Depending on how many terms x and y share, we have A 3, 4, 5, or 6. If A 3, then x y or x y 1, so that xx, yy xxy Z 3. If A 4 then x y, y 1, and xx, yy ď S A S 4, so that we are in the setting of part (a). So xx, yy A 4. Similarly, if A 5, then
3 we are in the second setting of part (b) and xx, yy A 5. If A 6, then x and y are disjoint, so that xy yx. So xx, yy Z 3 ˆ Z Group actions. (a) Let G ü A. Prove that if a, b P A and b g a for some g P G, then G b gg a g 1. Deduce that if G acts transitively on A, then the kernel of the action is Ş gpg gg ag 1. Proof. If b g a and x P G a, then pgxg 1 q b g px pg 1 g aqqq g px aq g a b so gg a g 1 Ď G b. Similarly, since b g a, we have a g 1 a, so that g 1 G b g Ď G a. This is true if and only if G b Ď gg a g 1. So G b gg a g 1. If G acts transitively on A, there is only one orbit, so that every b P A can be written as g b a for some g b P G. So ker č G b č g b G g 1 a b č gg g 1 a. bpa bpa gpg (b) Let S 3 act on the set of ordered triples A tpi, j, kq i, j, k P r3su. (i) Find the orbits of S 3 ü A. Answer. Note that for any σ P S n, i j if and only if σpiq σpjq (and similarly for i and k, and j and k). And for l P rns there is some σ P S n such that σpiq l (and similarly for j and k). So the orbits of the action on A are classified by which entries are the same and which aren t. For example, O p1,1,2q tpi, i, jq i, j P rns, i ju and O p1,2,1q tpi, j, iq i, j P rns, i ju. So there are five orbits in total, given by O p1,1,1q, O p1,1,2q, O p1,2,1q, O p2,1,1q, and O p1,2,3q. (ii) For each orbit O, choose one representative a P O and calculate G a. G : G a O. Verify that
4 4 Answer. a p1, 1, 1q : G a t1, p23qu, so G : G a 6{2 3 O a 3 a p1, 1, 2q : G a t1u, so G : G a 6{1 6 O a a p1, 2, 1q : G a t1u, so G : G a 6{1 6 O a a p2, 1, 1q : G a t1u, so G : G a 6{1 6 O a a p1, 2, 3q : G a t1u, so G : G a 6{1 6 O a (c) Briefly give an example of an action of a group G on a set A satisfying the following properties. (It s ok to give examples from class or the book.) (a) transitive and faithful Answer. G 1 acting on A tau trivially. (b) transitive but not faithful Answer. G Z 2 acting on A tau trivially. (c) faithful but not transitive Answer. G 1 acting on A ta, bu trivially. (d) not transitive or faithful Answer. G Z 2 acting on A ta, bu trivially. (e) transitive, faithful, and trivial Answer. G 1 acting on A tau trivially. (d) Suppose G acts transitively on a finite set A, and let H IJ G. Note that the action of G on A restricts to an action of H on A, which is not necessarily transitive anymore. Let O 1, O 2,..., O r be the distinct orbits of the action of H on A. (i) Show that for each a P A, H a G a X H.
5 5 Proof. We have H a th P H h a au tg P G g a au X H G a X H. (ii) Prove that G permutes O 1, O 2,..., O r, i.e. for each g P G, i P rrs, we hace g O i O j for some j P rrs (where g O i : tg a a P O i u); and σ g : to 1, O 2,..., O r u Ñ to 1, O 2,..., O r u defined by O i ÞÑ g O i is a bijection for each g P G. Proof. If we write O i H a i for some a i P O i, then since H is normal in G, we have ( ) g O i g ph a i q gh a i Hg a i H pg a i q. So since the orbits partition A and g a P A, we have that H pg aq is one of the orbits. And since left action by g 1 inverts the left action by g, σ g is a bijection. (iii) Deduce that G acts on the set A to 1, O 2,..., O r u. Show that this action is transitive, and deduce that O i O j for all i, j P rrs. Proof. Using ( ), we have 1 O i H p1 a i q H a i O i and g 1 pg 2 O i q g 1 H pg 2 a i q H pg 1 pg 2 a i qq H pg 1 g 2 a 1 q g 1 g 2 H a 1 g 1 g 2 O 1. So G acts on A. For all orbits H a, H b P A, since G acts transitively on A, there is some g P G such that b g a, so So the action on A is transitive. g ph aq H pg aq H b. Since the map from one orbit to the other given by O i ÞÑ g O i is a bijection (since it is invertible by left action of g 1 ) we have O i g O i for all g P G. But G acts transitively on A, so O i O j for all i, j P rrs. (iv) Fix O P A, and let a P O (so that O H a). Show that O H : H X G a and that r G : HG a (where r A as above). Proof. Since O H : H a and H a H X G a, it follows that O H : H X G a. Now let r A be the number of H-orbits. Since H IJ G, the second isomorphism theorem tells us that HG a ď G, and give us the following lattice, where the marked
6 6 edges (from subgroup A down to subgroup B) indicate the same index ( A : B ): H G HG a H X G a 1 G a So HG a : G a H : H X G a, and HG a : H G a : H X G a. And since, for any C ď B ď A, we have C : A A : B B : C. So G : G a G : HG a HG a : G a. Since the action of G is transitive, we have A G : G a. And since the H-orbits partition A, and every orbit has the same size, we have A r O for any O P A. So G : G a r O r H : H a r H : H X G a r HG a : G a. Therefore, since G : G a, r, and O are all finite, and G : G a G : HG a HG a : G a, we have r G : G a { HG a : G a G : HG a. 3. Left actions of groups on themselves (a) Label the elements of S 3 as label: x 1 x 2 x 3 x 4 x 5 x 6 element: 1 p12q p23q p13q p123q p132q Let S 3 act on itself by left multiplication (σ τ στ for σ, τ P S n ), and consider the maps ϕ : S 3 Ñ S S3 S tx1,...,x 6 u induced by the action of S 3 on itself, and ψ : S tx1,...,x 6 u Ñ S 6 defined by ψpσq : i ÞÑ j iff σ : x i ÞÑ x j. For example, since p12q x 1 p12q1 p12q x 2, p12q x 2 p12qp12q 1 x 1, p12q x 3 p12qp23q p123q x 5, p12q x 4 p12qp13q p132q x 6, p12q x 5 p12qp123q p23q x 3, p12q x 6 p12qp132q p13q x 4, we have ϕpp12qq is the map that sends x 1 ÞÑ x 2, x 2 ÞÑ x 1, x 3 ÞÑ x 5, x 4 ÞÑ x 6, x 5 ÞÑ x 3, x 6 ÞÑ x 4, and ψpϕpp12qqq is the map that sends 1 ÞÑ 2, 2 ÞÑ 1, 3 ÞÑ 5, 4 ÞÑ 6, 5 ÞÑ 3, 6 ÞÑ 4, so that ψpϕpp12qqq p12qp35qp46q. Give the images of the rest of the elements of S 3 under ψ ϕ, and verify that S 3 ψ ϕps 3 q (since ψ ϕ is injective). [It s ok to google multiplication table for S 3 to expedite this problem.]
7 Answer. We have the action of S 3 on tx 1,..., x 6 u is given by x 1 x 2 x 3 x 4 x 5 x 6 1 x 1 x 2 x 3 x 4 x 5 x 6 p12q x 2 x 1 x 5 x 6 x 3 x 4 p23q x 3 x 6 x 1 x 5 x 4 x 2 p13q x 4 x 5 x 6 x 1 x 2 x 3 p123q x 5 x 4 x 2 x 3 x 6 x 1 p132q x 6 x 3 x 4 x 2 x 1 x 5 so that ψ ϕ : 1 ÞÑ 1 p12q ÞÑ p12qp35qp46q p23q ÞÑ p13qp26qp45q p13q ÞÑ p14qp25qp36q p123q ÞÑ p156qp243q p132q ÞÑ p165qp234q Since ψ ϕ is a homomorphism and has 6 distinct elements in its image, we have S 3 ψ ϕps 3 q. (b) Consider H xsy inside of D 2 3, and label the cosets as label: x 1 x 2 x 3 element: H rh r 2 H. Repeat problem 1, listing the images of ψ ϕ : D 6 Ñ S 3 (where ϕ : D 6 Ñ S tgh gpd6 u S tx1,x 2,x 3 u and ψ : S tx1,x 2,x 3 u Ñ S 3 are defined analogously. Answer. We have H t1, su, rh tr, rsu tr, sr 2 u, r 2 H tr 2, r 2 su tr 2, sru. So the action of D 6 on tx 1, x 2, x 3 u is given by x 1 x 2 x 3 1 x 1 x 2 x 3 r x 2 x 3 x 1 r 2 x 3 x 1 x 2 s x 1 x 3 x 2 sr x 3 x 2 x 1 sr 2 x 2 x 1 x 3 so that ψ ϕ : 1 ÞÑ 1 r ÞÑ p123q r 2 ÞÑ p132q s ÞÑ p23q sr ÞÑ p13q sr 2 ÞÑ p12q Since ψ ϕ is a homomorphism and has 6 distinct elements in its image, we have D 6 ψ ϕpd 6 q S 3. (c) Show that if H has finite index n in G, then there is a normal subgroup K IJ G with K ď H and G : K ď n!. Proof. Let G act on the left cosets of H, of which there are n. This action corresponds to a homomorphism σ : G Ñ S A, where A tgh g P Gu. Let K kerpσq. We have So G : K σpgq ď S n n!. G{K σpgq ď S A S n. 7
For example, p12q p2x 1 x 2 ` 5x 2 x 2 3 q 2x 2 x 1 ` 5x 1 x 2 3. (a) Let p 12x 5 1x 7 2x 4 18x 6 2x 3 ` 11x 1 x 2 x 3 x 4,
SOLUTIONS Math A4900 Homework 5 10/4/2017 1. (DF 2.2.12(a)-(d)+) Symmetric polynomials. The group S n acts on the set tx 1, x 2,..., x n u by σ x i x σpiq. That action extends to a function S n ˆ A Ñ A,
More informationHOMEWORK 4 MATH B4900 DUE: 2/28/ Annihilators. Let R be a ring with 1, and let M be an R-module. The annihilator of M in R is
HOMEWORK 4 MATH B4900 DUE: 2/28/2018 SOLUTIONS Math B4900 Homework 4 2/28/2018 1. Annihilators. Let R be a ring with 1, and let M be an R-module. The annihilator of M in R is (a) Show that AnnpMq is an
More informationLast time: Recall that the fibers of a map ϕ : X Ñ Y are the sets in ϕ 1 pyq Ď X which all map to the same element y P Y.
Last time: Recall that the fibers of a map ϕ : X Ñ Y are the sets in ϕ 1 pyq Ď X which all map to the same element y P Y. Last time: Recall that the fibers of a map ϕ : X Ñ Y are the sets in ϕ 1 pyq Ď
More informationReview: Review: 'pgq imgp'q th P H h 'pgq for some g P Gu H; kerp'q tg P G 'pgq 1 H u G.
Review: A homomorphism is a map ' : G Ñ H between groups satisfying 'pg 1 g 2 q 'pg 1 q'pg 2 q for all g 1,g 2 P G. Anisomorphism is homomorphism that is also a bijection. We showed that for any homomorphism
More informationWarmup Recall, a group H is cyclic if H can be generated by a single element. In other words, there is some element x P H for which Multiplicative
Warmup Recall, a group H is cyclic if H can be generated by a single element. In other words, there is some element x P H for which Multiplicative notation: H tx l l P Zu xxy, Additive notation: H tlx
More informationA TASTE OF COMBINATORIAL REPRESENTATION THEORY. MATH B4900 5/02/2018
A TASTE OF COMBINATORIAL REPRESENTATION THEORY. MATH B4900 5/02/2018 Young s Lattice is an infinite leveled labeled graph with vertices and edges as follows. Vertices: Label vertices in label vertices
More informationDr. Marques Sophie Algebra 1 Spring Semester 2017 Problem Set 9
Dr. Marques Sophie Algebra Spring Semester 207 Office 59 marques@cims.nyu.edu Problem Set 9 Exercise 0 : Prove that every group of order G 28 must contain a normal subgroup of order 7. (Why should it contain
More informationGroups and Symmetries
Groups and Symmetries Definition: Symmetry A symmetry of a shape is a rigid motion that takes vertices to vertices, edges to edges. Note: A rigid motion preserves angles and distances. Definition: Group
More informationLast time: isomorphism theorems. Let G be a group. 1. If ϕ : G Ñ H is a homomorphism of groups, then kerpϕq IJ G and
Last time: isomorphism theorems Let G be a group. 1. If ϕ : G Ñ H is a homomorphism of groups, then kerpϕq IJ G and G{kerpϕq ϕpgq. Last time: isomorphism theorems Let G be a group. 1. If ϕ : G Ñ H is a
More informationTeddy Einstein Math 4320
Teddy Einstein Math 4320 HW4 Solutions Problem 1: 2.92 An automorphism of a group G is an isomorphism G G. i. Prove that Aut G is a group under composition. Proof. Let f, g Aut G. Then f g is a bijective
More informationSchool of Mathematics and Statistics. MT5824 Topics in Groups. Problem Sheet I: Revision and Re-Activation
MRQ 2009 School of Mathematics and Statistics MT5824 Topics in Groups Problem Sheet I: Revision and Re-Activation 1. Let H and K be subgroups of a group G. Define HK = {hk h H, k K }. (a) Show that HK
More informationSolutions to Assignment 4
1. Let G be a finite, abelian group written additively. Let x = g G g, and let G 2 be the subgroup of G defined by G 2 = {g G 2g = 0}. (a) Show that x = g G 2 g. (b) Show that x = 0 if G 2 = 2. If G 2
More informationSUPPLEMENT ON THE SYMMETRIC GROUP
SUPPLEMENT ON THE SYMMETRIC GROUP RUSS WOODROOFE I presented a couple of aspects of the theory of the symmetric group S n differently than what is in Herstein. These notes will sketch this material. You
More information(5.11) (Second Isomorphism Theorem) If K G and N G, then K/(N K) = NK/N. PF: Verify N HK. Find a homomorphism f : K HK/N with ker(f) = (N K).
Lecture Note of Week 3 6. Normality, Quotients and Homomorphisms (5.7) A subgroup N satisfying any one properties of (5.6) is called a normal subgroup of G. Denote this fact by N G. The homomorphism π
More informationLecture 24 Properties of deals
Lecture 24 Properties of deals Aside: Representation theory of finite groups Let G be a finite group, and let R C, R, or Q (any commutative ring). Aside: Representation theory of finite groups Let G be
More informationMath 594, HW2 - Solutions
Math 594, HW2 - Solutions Gilad Pagi, Feng Zhu February 8, 2015 1 a). It suffices to check that NA is closed under the group operation, and contains identities and inverses: NA is closed under the group
More informationExercises on chapter 1
Exercises on chapter 1 1. Let G be a group and H and K be subgroups. Let HK = {hk h H, k K}. (i) Prove that HK is a subgroup of G if and only if HK = KH. (ii) If either H or K is a normal subgroup of G
More informationMath 210A: Algebra, Homework 5
Math 210A: Algebra, Homework 5 Ian Coley November 5, 2013 Problem 1. Prove that two elements σ and τ in S n are conjugate if and only if type σ = type τ. Suppose first that σ and τ are cycles. Suppose
More informationD-MATH Algebra I HS 2013 Prof. Brent Doran. Solution 3. Modular arithmetic, quotients, product groups
D-MATH Algebra I HS 2013 Prof. Brent Doran Solution 3 Modular arithmetic, quotients, product groups 1. Show that the functions f = 1/x, g = (x 1)/x generate a group of functions, the law of composition
More informationProblem Set #7. Exercise 1 : In GLpn, Cq and SLpn, Cq define the subgroups of scalar matrices
Dr. Marques Sophie Algebra Spring Semester 207 Office 59 marques@cims.nyu.edu Problem Set #7 In the following, denotes a isomorphism of groups. Exercise 0 : Let G be a finite abelian group with G n, we
More informationFall /29/18 Time Limit: 75 Minutes
Math 411: Abstract Algebra Fall 2018 Midterm 10/29/18 Time Limit: 75 Minutes Name (Print): Solutions JHU-ID: This exam contains 8 pages (including this cover page) and 6 problems. Check to see if any pages
More informationIntroduction to Groups
Introduction to Groups Hong-Jian Lai August 2000 1. Basic Concepts and Facts (1.1) A semigroup is an ordered pair (G, ) where G is a nonempty set and is a binary operation on G satisfying: (G1) a (b c)
More informationHW2 Solutions Problem 1: 2.22 Find the sign and inverse of the permutation shown in the book (and below).
Teddy Einstein Math 430 HW Solutions Problem 1:. Find the sign and inverse of the permutation shown in the book (and below). Proof. Its disjoint cycle decomposition is: (19)(8)(37)(46) which immediately
More informationCONSEQUENCES OF THE SYLOW THEOREMS
CONSEQUENCES OF THE SYLOW THEOREMS KEITH CONRAD For a group theorist, Sylow s Theorem is such a basic tool, and so fundamental, that it is used almost without thinking, like breathing. Geoff Robinson 1.
More informationDefinitions, Theorems and Exercises. Abstract Algebra Math 332. Ethan D. Bloch
Definitions, Theorems and Exercises Abstract Algebra Math 332 Ethan D. Bloch December 26, 2013 ii Contents 1 Binary Operations 3 1.1 Binary Operations............................... 4 1.2 Isomorphic Binary
More informationGROUP ACTIONS EMMANUEL KOWALSKI
GROUP ACTIONS EMMANUEL KOWALSKI Definition 1. Let G be a group and T a set. An action of G on T is a map a: G T T, that we denote a(g, t) = g t, such that (1) For all t T, we have e G t = t. (2) For all
More informationSome practice problems for midterm 2
Some practice problems for midterm 2 Kiumars Kaveh November 14, 2011 Problem: Let Z = {a G ax = xa, x G} be the center of a group G. Prove that Z is a normal subgroup of G. Solution: First we prove Z is
More informationMATH 101: ALGEBRA I WORKSHEET, DAY #3. Fill in the blanks as we finish our first pass on prerequisites of group theory.
MATH 101: ALGEBRA I WORKSHEET, DAY #3 Fill in the blanks as we finish our first pass on prerequisites of group theory 1 Subgroups, cosets Let G be a group Recall that a subgroup H G is a subset that is
More information120A LECTURE OUTLINES
120A LECTURE OUTLINES RUI WANG CONTENTS 1. Lecture 1. Introduction 1 2 1.1. An algebraic object to study 2 1.2. Group 2 1.3. Isomorphic binary operations 2 2. Lecture 2. Introduction 2 3 2.1. The multiplication
More informationDISCRETE MATH (A LITTLE) & BASIC GROUP THEORY - PART 3/3. Contents
DISCRETE MATH (A LITTLE) & BASIC GROUP THEORY - PART 3/3 T.K.SUBRAHMONIAN MOOTHATHU Contents 1. Cayley s Theorem 1 2. The permutation group S n 2 3. Center of a group, and centralizers 4 4. Group actions
More informationSOLUTIONS Math B4900 Homework 9 4/18/2018
SOLUTIONS Math B4900 Homework 9 4/18/2018 1. Show that if G is a finite group and F is a field, then any simple F G-modules is finitedimensional. [This is not a consequence of Maschke s theorem; it s just
More informationGroup Actions Definition. Let G be a group, and let X be a set. A left action of G on X is a function θ : G X X satisfying:
Group Actions 8-26-202 Definition. Let G be a group, and let X be a set. A left action of G on X is a function θ : G X X satisfying: (a) θ(g,θ(g 2,x)) = θ(g g 2,x) for all g,g 2 G and x X. (b) θ(,x) =
More information1 Chapter 6 - Exercise 1.8.cf
1 CHAPTER 6 - EXERCISE 1.8.CF 1 1 Chapter 6 - Exercise 1.8.cf Determine 1 The Class Equation of the dihedral group D 5. Note first that D 5 = 10 = 5 2. Hence every conjugacy class will have order 1, 2
More informationLast time: Cyclic groups. Proposition
Warmup Recall, a group H is cyclic if H can be generated by a single element. In other words, there is some element x P H for which Multiplicative notation: H tx` ` P Zu xxy, Additive notation: H t`x `
More information1. Examples. We did most of the following in class in passing. Now compile all that data.
SOLUTIONS Math A4900 Homework 12 11/22/2017 1. Examples. We did most of the following in class in passing. Now compile all that data. (a) Favorite examples: Let R tr, Z, Z{3Z, Z{6Z, M 2 prq, Rrxs, Zrxs,
More informationHOMEWORK 3 LOUIS-PHILIPPE THIBAULT
HOMEWORK 3 LOUIS-PHILIPPE THIBAULT Problem 1 Let G be a group of order 56. We have that 56 = 2 3 7. Then, using Sylow s theorem, we have that the only possibilities for the number of Sylow-p subgroups
More informationMath 122 Midterm 2 Fall 2014 Solutions
Math 122 Midterm 2 Fall 2014 Solutions Common mistakes i. Groups of order pq are not always cyclic. Look back on Homework Eight. Also consider the dihedral groups D 2n for n an odd prime. ii. If H G and
More informationGroup Theory
Group Theory 2014 2015 Solutions to the exam of 4 November 2014 13 November 2014 Question 1 (a) For every number n in the set {1, 2,..., 2013} there is exactly one transposition (n n + 1) in σ, so σ is
More informationDefinitions. Notations. Injective, Surjective and Bijective. Divides. Cartesian Product. Relations. Equivalence Relations
Page 1 Definitions Tuesday, May 8, 2018 12:23 AM Notations " " means "equals, by definition" the set of all real numbers the set of integers Denote a function from a set to a set by Denote the image of
More informationETIKA V PROFESII PSYCHOLÓGA
P r a ž s k á v y s o k á š k o l a p s y c h o s o c i á l n í c h s t u d i í ETIKA V PROFESII PSYCHOLÓGA N a t á l i a S l o b o d n í k o v á v e d ú c i p r á c e : P h D r. M a r t i n S t r o u
More informationElements of solution for Homework 5
Elements of solution for Homework 5 General remarks How to use the First Isomorphism Theorem A standard way to prove statements of the form G/H is isomorphic to Γ is to construct a homomorphism ϕ : G Γ
More informationMath 451, 01, Exam #2 Answer Key
Math 451, 01, Exam #2 Answer Key 1. (25 points): If the statement is always true, circle True and prove it. If the statement is never true, circle False and prove that it can never be true. If the statement
More informationMA441: Algebraic Structures I. Lecture 26
MA441: Algebraic Structures I Lecture 26 10 December 2003 1 (page 179) Example 13: A 4 has no subgroup of order 6. BWOC, suppose H < A 4 has order 6. Then H A 4, since it has index 2. Thus A 4 /H has order
More informationZachary Scherr Math 370 HW 7 Solutions
1 Book Problems 1. 2.7.4b Solution: Let U 1 {u 1 u U} and let S U U 1. Then (U) is the set of all elements of G which are finite products of elements of S. We are told that for any u U and g G we have
More informationISOMORPHISMS KEITH CONRAD
ISOMORPHISMS KEITH CONRAD 1. Introduction Groups that are not literally the same may be structurally the same. An example of this idea from high school math is the relation between multiplication and addition
More informationMath 120: Homework 2 Solutions
Math 120: Homework 2 Solutions October 12, 2018 Problem 1.2 # 9. Let G be the group of rigid motions of the tetrahedron. Show that G = 12. Solution. Let us label the vertices of the tetrahedron 1, 2, 3,
More informationA DARK GREY P O N T, with a Switch Tail, and a small Star on the Forehead. Any
Y Y Y X X «/ YY Y Y ««Y x ) & \ & & } # Y \#$& / Y Y X» \\ / X X X x & Y Y X «q «z \x» = q Y # % \ & [ & Z \ & { + % ) / / «q zy» / & / / / & x x X / % % ) Y x X Y $ Z % Y Y x x } / % «] «] # z» & Y X»
More informationEXERCISES ON THE OUTER AUTOMORPHISMS OF S 6
EXERCISES ON THE OUTER AUTOMORPHISMS OF S 6 AARON LANDESMAN 1. INTRODUCTION In this class, we investigate the outer automorphism of S 6. Let s recall some definitions, so that we can state what an outer
More informationABSTRACT ALGEBRA 1, LECTURES NOTES 5: SUBGROUPS, CONJUGACY, NORMALITY, QUOTIENT GROUPS, AND EXTENSIONS.
ABSTRACT ALGEBRA 1, LECTURES NOTES 5: SUBGROUPS, CONJUGACY, NORMALITY, QUOTIENT GROUPS, AND EXTENSIONS. ANDREW SALCH 1. Subgroups, conjugacy, normality. I think you already know what a subgroup is: Definition
More informationn ) = f (x 1 ) e 1... f (x n ) e n
1. FREE GROUPS AND PRESENTATIONS Let X be a subset of a group G. The subgroup generated by X, denoted X, is the intersection of all subgroups of G containing X as a subset. If g G, then g X g can be written
More informationMath 121 Homework 5: Notes on Selected Problems
Math 121 Homework 5: Notes on Selected Problems 12.1.2. Let M be a module over the integral domain R. (a) Assume that M has rank n and that x 1,..., x n is any maximal set of linearly independent elements
More informationAnswers to Final Exam
Answers to Final Exam MA441: Algebraic Structures I 20 December 2003 1) Definitions (20 points) 1. Given a subgroup H G, define the quotient group G/H. (Describe the set and the group operation.) The quotient
More informationHonors Algebra 4, MATH 371 Winter 2010 Assignment 3 Due Friday, February 5 at 08:35
Honors Algebra 4, MATH 371 Winter 2010 Assignment 3 Due Friday, February 5 at 08:35 1. Let R 0 be a commutative ring with 1 and let S R be the subset of nonzero elements which are not zero divisors. (a)
More informationHomework Problems, Math 200, Fall 2011 (Robert Boltje)
Homework Problems, Math 200, Fall 2011 (Robert Boltje) Due Friday, September 30: ( ) 0 a 1. Let S be the set of all matrices with entries a, b Z. Show 0 b that S is a semigroup under matrix multiplication
More informationAPPROXIMATE HOMOMORPHISMS BETWEEN THE BOOLEAN CUBE AND GROUPS OF PRIME ORDER
APPROXIMATE HOMOMORPHISMS BETWEEN THE BOOLEAN CUBE AND GROUPS OF PRIME ORDER TOM SANDERS The purpose of this note is to highlight a question raised by Shachar Lovett [Lov], and to offer some motivation
More information3. G. Groups, as men, will be known by their actions. - Guillermo Moreno
3.1. The denition. 3. G Groups, as men, will be known by their actions. - Guillermo Moreno D 3.1. An action of a group G on a set X is a function from : G X! X such that the following hold for all g, h
More informationMODEL ANSWERS TO HWK #4. ϕ(ab) = [ab] = [a][b]
MODEL ANSWERS TO HWK #4 1. (i) Yes. Given a and b Z, ϕ(ab) = [ab] = [a][b] = ϕ(a)ϕ(b). This map is clearly surjective but not injective. Indeed the kernel is easily seen to be nz. (ii) No. Suppose that
More informationBASIC GROUP THEORY : G G G,
BASIC GROUP THEORY 18.904 1. Definitions Definition 1.1. A group (G, ) is a set G with a binary operation : G G G, and a unit e G, possessing the following properties. (1) Unital: for g G, we have g e
More informationBasic Definitions: Group, subgroup, order of a group, order of an element, Abelian, center, centralizer, identity, inverse, closed.
Math 546 Review Exam 2 NOTE: An (*) at the end of a line indicates that you will not be asked for the proof of that specific item on the exam But you should still understand the idea and be able to apply
More informationChapter 9: Group actions
Chapter 9: Group actions Matthew Macauley Department of Mathematical Sciences Clemson University http://www.math.clemson.edu/~macaule/ Math 4120, Summer I 2014 M. Macauley (Clemson) Chapter 9: Group actions
More information(Think: three copies of C) i j = k = j i, j k = i = k j, k i = j = i k.
Warm-up: The quaternion group, denoted Q 8, is the set {1, 1, i, i, j, j, k, k} with product given by 1 a = a 1 = a a Q 8, ( 1) ( 1) = 1, i 2 = j 2 = k 2 = 1, ( 1) a = a ( 1) = a a Q 8, (Think: three copies
More informationMODEL ANSWERS TO THE FIFTH HOMEWORK
MODEL ANSWERS TO THE FIFTH HOMEWORK 1. Chapter 3, Section 5: 1 (a) Yes. Given a and b Z, φ(ab) = [ab] = [a][b] = φ(a)φ(b). This map is clearly surjective but not injective. Indeed the kernel is easily
More informationSF2729 GROUPS AND RINGS LECTURE NOTES
SF2729 GROUPS AND RINGS LECTURE NOTES 2011-03-01 MATS BOIJ 6. THE SIXTH LECTURE - GROUP ACTIONS In the sixth lecture we study what happens when groups acts on sets. 1 Recall that we have already when looking
More informationFall 2014 Math 122 Midterm 1
1. Some things you ve (maybe) done before. 5 points each. (a) If g and h are elements of a group G, show that (gh) 1 = h 1 g 1. (gh)(h 1 g 1 )=g(hh 1 )g 1 = g1g 1 = gg 1 =1. Likewise, (h 1 g 1 )(gh) =h
More informationGALOIS GROUPS AS PERMUTATION GROUPS
GALOIS GROUPS AS PERMUTATION GROUPS KEITH CONRAD 1. Introduction A Galois group is a group of field automorphisms under composition. By looking at the effect of a Galois group on field generators we can
More informationPRINCIPLES OF ANALYSIS - LECTURE NOTES
PRINCIPLES OF ANALYSIS - LECTURE NOTES PETER A. PERRY 1. Constructions of Z, Q, R Beginning with the natural numbers N t1, 2, 3,...u we can use set theory to construct, successively, Z, Q, and R. We ll
More informationExam 1 Solutions. Solution: The 16 contributes 5 to the total and contributes 2. All totaled, there are 5 ˆ 2 10 abelian groups.
Math 5372 Spring 2014 Exam 1 Solutions 1. (15 points) How many abelian groups are there of order: (a) 16 For any prime p, there are as many groups of order p k as there are partitions of k. The number
More informationMath 4400, Spring 08, Sample problems Final Exam.
Math 4400, Spring 08, Sample problems Final Exam. 1. Groups (1) (a) Let a be an element of a group G. Define the notions of exponent of a and period of a. (b) Suppose a has a finite period. Prove that
More informationEXERCISE SHEET 1 WITH SOLUTIONS
EXERCISE SHEET 1 WITH SOLUTIONS (E8) Prove that, given a transitive action of G on Ω, there exists a subgroup H G such that the action of G on Ω is isomorphic to the action of G on H\G. You may need to
More informationNotes on Group Theory. by Avinash Sathaye, Professor of Mathematics November 5, 2013
Notes on Group Theory by Avinash Sathaye, Professor of Mathematics November 5, 2013 Contents 1 Preparation. 2 2 Group axioms and definitions. 2 Shortcuts................................. 2 2.1 Cyclic groups............................
More informationAlgebra-I, Fall Solutions to Midterm #1
Algebra-I, Fall 2018. Solutions to Midterm #1 1. Let G be a group, H, K subgroups of G and a, b G. (a) (6 pts) Suppose that ah = bk. Prove that H = K. Solution: (a) Multiplying both sides by b 1 on the
More informationAlgebra homework 6 Homomorphisms, isomorphisms
MATH-UA.343.005 T.A. Louis Guigo Algebra homework 6 Homomorphisms, isomorphisms Exercise 1. Show that the following maps are group homomorphisms and compute their kernels. (a f : (R, (GL 2 (R, given by
More informationOn Maximal Subgroups of a Group with Unique Order
Available online at wwwscholarsresearchlibrarycom European Journal of Applied Engineering and Scientific Research, 208, 6():2-3 On Maximal Subgroups of a Group with Unique Order ISSN: 2278-004 M Bello
More informationMath 581 Problem Set 3 Solutions
Math 581 Problem Set 3 Solutions 1. Prove that complex conjugation is a isomorphism from C to C. Proof: First we prove that it is a homomorphism. Define : C C by (z) = z. Note that (1) = 1. The other properties
More informationProblem 1. Let I and J be ideals in a ring commutative ring R with 1 R. Recall
I. Take-Home Portion: Math 350 Final Exam Due by 5:00pm on Tues. 5/12/15 No resources/devices other than our class textbook and class notes/handouts may be used. You must work alone. Choose any 5 problems
More informationMATH HL OPTION - REVISION SETS, RELATIONS AND GROUPS Compiled by: Christos Nikolaidis
MATH HL OPTION - REVISION SETS, RELATIONS AND GROUPS Compiled by: Christos Nikolaidis PART B: GROUPS GROUPS 1. ab The binary operation a * b is defined by a * b = a+ b +. (a) Prove that * is associative.
More informationCourse 311: Abstract Algebra Academic year
Course 311: Abstract Algebra Academic year 2007-08 D. R. Wilkins Copyright c David R. Wilkins 1997 2007 Contents 1 Topics in Group Theory 1 1.1 Groups............................... 1 1.2 Examples of Groups.......................
More informationMath 2070BC Term 2 Weeks 1 13 Lecture Notes
Math 2070BC 2017 18 Term 2 Weeks 1 13 Lecture Notes Keywords: group operation multiplication associative identity element inverse commutative abelian group Special Linear Group order infinite order cyclic
More informationGroups. Chapter The symmetric group
Chapter 1 Groups In this chapter we ll cover pretty much all of group theory. This material is roughly the same as Rotman s chapters 2 and 5, but beware there are some extra things not in Rotman... You
More informationif G permutes the set of its subgroups by conjugation then Stab G (H) = N G (H),
1 0.1 G-sets We introduce a part of the theory of G-sets, suitable for understanding the approach GAP uses to compute with permutation groups, using stabilizer chains. Rotman s book describes other results
More informationThe Symmetric Groups
Chapter 7 The Symmetric Groups 7. Introduction In the investigation of finite groups the symmetric groups play an important role. Often we are able to achieve a better understanding of a group if we can
More informationDIHEDRAL GROUPS II KEITH CONRAD
DIHEDRAL GROUPS II KEITH CONRAD We will characterize dihedral groups in terms of generators and relations, and describe the subgroups of D n, including the normal subgroups. We will also introduce an infinite
More informationMath 4320, Spring 2011
Math 4320, Spring 2011 Prelim 2 with solutions 1. For n =16, 17, 18, 19 or 20, express Z n (A product can have one or more factors.) as a product of cyclic groups. Solution. For n = 16, G = Z n = {[1],
More informationMath 429/581 (Advanced) Group Theory. Summary of Definitions, Examples, and Theorems by Stefan Gille
Math 429/581 (Advanced) Group Theory Summary of Definitions, Examples, and Theorems by Stefan Gille 1 2 0. Group Operations 0.1. Definition. Let G be a group and X a set. A (left) operation of G on X is
More informationA. (Groups of order 8.) (a) Which of the five groups G (as specified in the question) have the following property: G has a normal subgroup N such that
MATH 402A - Solutions for the suggested problems. A. (Groups of order 8. (a Which of the five groups G (as specified in the question have the following property: G has a normal subgroup N such that N =
More informationLecture 7 Cyclic groups and subgroups
Lecture 7 Cyclic groups and subgroups Review Types of groups we know Numbers: Z, Q, R, C, Q, R, C Matrices: (M n (F ), +), GL n (F ), where F = Q, R, or C. Modular groups: Z/nZ and (Z/nZ) Dihedral groups:
More informationCOMBINATORIAL GROUP THEORY NOTES
COMBINATORIAL GROUP THEORY NOTES These are being written as a companion to Chapter 1 of Hatcher. The aim is to give a description of some of the group theory required to work with the fundamental groups
More informationCrypto math II. Alin Tomescu May 27, Abstract A quick overview on group theory from Ron Rivest s course in Spring 2015.
Crypto math II Alin Tomescu alinush@mit.edu May 7, 015 Abstract A quick overview on group theory from Ron Rivest s 6.857 course in Spring 015. 1 Overview Group theory review Diffie-Hellman (DH) key exchange
More informationExercises MAT2200 spring 2013 Ark 4 Homomorphisms and factor groups
Exercises MAT2200 spring 2013 Ark 4 Homomorphisms and factor groups This Ark concerns the weeks No. (Mar ) and No. (Mar ). Plans until Eastern vacations: In the book the group theory included in the curriculum
More informationThe Outer Automorphism of S 6
Meena Jagadeesan 1 Karthik Karnik 2 Mentor: Akhil Mathew 1 Phillips Exeter Academy 2 Massachusetts Academy of Math and Science PRIMES Conference, May 2016 What is a Group? A group G is a set of elements
More informationChapter 16 MSM2P2 Symmetry And Groups
Chapter 16 MSM2P2 Symmetry And Groups 16.1 Symmetry 16.1.1 Symmetries Of The Square Definition 1 A symmetry is a function from an object to itself such that for any two points a and b in the object, the
More information(d) Since we can think of isometries of a regular 2n-gon as invertible linear operators on R 2, we get a 2-dimensional representation of G for
Solutions to Homework #7 0. Prove that [S n, S n ] = A n for every n 2 (where A n is the alternating group). Solution: Since [f, g] = f 1 g 1 fg is an even permutation for all f, g S n and since A n is
More informationSUMMARY ALGEBRA I LOUIS-PHILIPPE THIBAULT
SUMMARY ALGEBRA I LOUIS-PHILIPPE THIBAULT Contents 1. Group Theory 1 1.1. Basic Notions 1 1.2. Isomorphism Theorems 2 1.3. Jordan- Holder Theorem 2 1.4. Symmetric Group 3 1.5. Group action on Sets 3 1.6.
More information1.5 Applications Of The Sylow Theorems
14 CHAPTER1. GROUP THEORY 8. The Sylow theorems are about subgroups whose order is a power of a prime p. Here is a result about subgroups of index p. Let H be a subgroup of the finite group G, and assume
More informationSelected Solutions to Math 4107, Set 4
Selected Solutions to Math 4107, Set 4 November 9, 2005 Page 90. 1. There are 3 different classes: {e}, {(1 2), (1 3), (2 3)}, {(1 2 3), (1 3 2)}. We have that c e = 1, c (1 2) = 3, and c (1 2 3) = 2.
More informationMAT1100HF ALGEBRA: ASSIGNMENT II. Contents 1. Problem Problem Problem Problem Problem Problem
MAT1100HF ALEBRA: ASSINMENT II J.A. MRACEK 998055704 DEPARTMENT OF MATHEMATICS UNIVERSITY OF TORONTO Contents 1. Problem 1 1 2. Problem 2 2 3. Problem 3 2 4. Problem 4 3 5. Problem 5 3 6. Problem 6 3 7.
More informationRAPHAËL ROUQUIER. k( )
GLUING p-permutation MODULES 1. Introduction We give a local construction of the stable category of p-permutation modules : a p- permutation kg-module gives rise, via the Brauer functor, to a family of
More informationQuiz 2 Practice Problems
Quiz 2 Practice Problems Math 332, Spring 2010 Isomorphisms and Automorphisms 1. Let C be the group of complex numbers under the operation of addition, and define a function ϕ: C C by ϕ(a + bi) = a bi.
More informationits image and kernel. A subgroup of a group G is a non-empty subset K of G such that k 1 k 1
10 Chapter 1 Groups 1.1 Isomorphism theorems Throughout the chapter, we ll be studying the category of groups. Let G, H be groups. Recall that a homomorphism f : G H means a function such that f(g 1 g
More informationMATH 3005 ABSTRACT ALGEBRA I FINAL SOLUTION
MATH 3005 ABSTRACT ALGEBRA I FINAL SOLUTION SPRING 2014 - MOON Write your answer neatly and show steps. Any electronic devices including calculators, cell phones are not allowed. (1) Write the definition.
More information